Performance Prediction of Parallel Low-level Image Processing Operations

Transcription

1 Prformanc Prdiction of Paralll Low-lvl Imag Procssing Orations Zoltan Juhasz Dt. of Information Tchnology and Automation Univrsity of Vszrm Vszrm, P.O.Box 158, H-8201, Hungary hon: , fax: Abstract. An analytical rformanc rdiction mthod is rsntd for rdicting th rformanc of transutr basd low-lvl imag rocssing orations. Th mthod is basd on th fixd-sizd sdu formula and uss a systm ovrhad function to rrsnt th communication and comutation ovrhad as wll as th hardwar and imlmntation dndnt (.g. toology) charactristics of th aralll systm. To dtrmin th ffct of communication on rformanc, first th rortis of th transutr link ar invstigatd to crat a siml link modl, thn th communication attrns of th data-aralll algorithms ar analysd to driv th ovrhad function. Using this ovrhad function, th aralll xcution tim, sdu and scalability of a givn algorithm can b rdictd in advanc, without systm building, hling dvlors to dsign fficint and scalabl transutr-basd aralll rograms. 1. Introduction With th raid advancmnt of hardwar tchnology, massivly aralll comutrs ar bcoming mor and mor widly usd. Unfortunatly, th utilisation of ths machins, i.. th actual dlivrd comutational rformanc on ral roblms, is oftn only a fraction of th machins advrtisd ak rformanc [1]. This is mainly du to th infficint aralllisation of th roblms. Finding th bst aralll algorithm for a givn architctur is a comlx task. Unlik squntial rograms, whr th rformanc can b asily dtrmind by th rocssor sd and th numbr of rquird machin instructions, th rformanc of aralll algorithms dnds on svral othr factors, such as grain siz, communication sd, intrconnction toology, rocss allocation, tc., which maks th dsign has inhrntly mor difficult. An fficint rformanc rdiction mthod nabls th rogrammr to invstigat th ffct of aramtr and algorithm changs on th rformanc. It also hls to dtrmin what hardwar latform is bst suitd to a givn aralll algorithm and rovids information to hardwar architcts on what systm aramtrs nd to b changd to achiv imrovd rformanc. Paralll softwar is xctd to b scalabl, i.. th rogram should run on a varity of architcturs (from a fw-rocssor small scal systm to a massivly aralll machin consisting of thousands of rocssors) without any modification. This rorty nabls cost-ffctiv rototying on a small systm, and imrovs maintainability and ortability.

2 A good rformanc rdiction mthod should b siml, accurat and quick to b incororatd into th dsign has of th dvlomnt, othrwis xrimnting bcoms too costly. Th most frquntly followd mthod is th masur-and-modify aroach. In this itrativ rocss, th xcution tim of th imlmntd algorithm is masurd and if ncssary, changs ar mad to (hofully) imrov rformanc. This mthod, although siml, dos not rovid insight into th dtails of th imlmntation and it rquirs th us of th full-scal aralll systm. Simulation mthods ar somtims usd to rdict th rformanc of aralll rograms but thy ar vry tim-consuming and for larg roblms thy bcom too comlx to b fasibl [2]. Analytical mthods us modls of th architctur and th algorithm, and xrss th xcution tim in a closd form xrssion [1][3-5]. Thy allow xamining th ffct of systm aramtr changs and rdicting rformanc using only a fw-rocssor systm. Accurat rdiction is scially imortant in imag rocssing alications, whr oftn svr tim constraints must b mt. For achiving fast xcution, many rocssors must b mloyd, thrfor it is vital to know th rformanc bfor urchasing or building an xnsiv massivly aralll comutr. Furthrmor, as imag rocssing algorithms ar alid on fixd siz imags, roblms cannot b scald u to achiv highr rformanc on largr machins. Thrfor it is imortant to analys th givn imlmntation of th algorithm in th initial dvlomnt has to roduc an fficint and scalabl aralll algorithm. This ar rsnts an analytical rformanc rdiction mthod that can b usd to rdict th xcution tim and othr rformanc mtrics of low-lvl imag rocssing orations running on a rocssor aralll systm. Th assumd architcturs ar a ondimnsional and a two-dimnsional array of transutrs, connctd in a static narstnighbour attrn. Th most imortant art of th mthod is th calculation of th ovrhad function, as it is th ovrhad that dtrmins th achivabl sdu. Bcaus of th array toology, communication has a major ffct on th ovrhad. Th communication ovrhad, thrfor, is xamind carfully. Th mthod is illustratd on thr imag rocssing orations, th addition, convolution and histogram calculation, which rrsnt thr diffrnt classs of algorithms, th orations with no, local and global communication, rsctivly. Th dvlod analytical modl rovids an accurat and fast rdiction of rformanc. It rquirs only a fw systm aramtrs that can b obtaind from hardwar scification or from masurmnts on a two-transutr systm. Th mthod can hl rogrammrs in rformanc dbugging to dtct critical arts of an imlmntation and systm dsignrs to rdict rformanc as wll as th ffcts of changing hardwar aramtrs. Th organisation of th ar is th following. Sction 2 dvlos th analytical rformanc rdiction mthod and dscribs how it can b usd to imrov and analys rformanc. In Sction 3 th transutr communication mdium, th link is analysd for its rformanc in ordr to build a siml data transfr modl dscribing link bhaviour. Th ffct of rotocol choic, data siz and mssag startu tim on th communication latncy is xamind. Sction 4 invstigats th diffrnt communication attrns found in dataaralll, low-lvl imag rocssing algorithms and dtrmins th communication ovrhad. Combining th hardwar-lvl link modl, th communication and comutation ovrhad formulas, th systm ovrhad functions for th slctd orations ar drivd. Sction 5 rsnts and comars th rdictd and xrimntal rformanc rsults. Th accuracy of th analytical mthod is shown. Critical arts in th algorithm-machin combination ar highlightd and imrovmnts ar mad in ordr to achiv bttr rformanc. Finally, in Sction 6 conclusions ar mad and dirctions for futur work ar highlightd.

3 2. Dvloing th rformanc modl Th main motivation for dvloing th following analytical modl was to rovid a tool for th rogrammr to rdict th rformanc of a givn imlmntation of a transutr-basd mssag-assing aralll rogram rior to imlmntation. Th modl should rovid an accurat rdiction of xcution tim and sdu and mak it ossibl to study th ffcts of systm aramtr changs. Svral rformanc mtrics ar usd for rformanc rdiction. Our modl is basd on th roblm siz bound sdu and is somwhat influncd by th mthod of isofficincy analysis [3]. Unlik isofficincy analysis, howvr, our mthod uss xact rdiction of xcution tim and sdu, not only asymtotic analysis. W dfin aralll xcution tim of an algorithm as T = T 1 / + T o () whr T 1 is th xcution tim of th srial vrsion of th algorithm on on rocssor, is th numbr of rocssors in th systm and T o () is th ovrhad function that rrsnts communication ovrhad, synchronisation dlays and th xcution of unaralllisabl arts of th algorithm. Sdu is dfind as S() = T 1 /T. Substituting th xrssion for aralll xcution tim into th sdu formula, w arriv at S b g = =, (1) + To ( ) + To ( ) 1 1 T 1 whr W is th roblm siz (th numbr of orations to b rformd) and t c is th unit cost of an oration. Th focus of th analysis is on th trm T o ()/Wt c, sinc this trm will dtrmin th ovrall rformanc. To achiv idal rformanc and scalability, this trm should b invrs roortional to. If this is not ossibl, th ovrhad should b constant or in th worst cas, a slowly incrasing function of. Onc th ovrhad function is known, th rquird rformanc mtric can b calculatd. Th valu of T 1 can b obtaind from a trial run or can b calculatd from rocssor scifications. Th modl can b thought of as a layrd systm [6], whr th hardwar-layr is rrsntd by t c and by th communication systm aramtrs in T o (). Th aralll and alication layrs ar rrsntd by th ovrhad function, which xrsss th communication attrn (consquntly th aralll dcomosition) of th algorithm. Wt c 3. Analysing link rformanc To dscrib th ffct of communication on th ovrhad accuratly, it is ncssary to xamin th transutr link and dtrmin how th choic of rotocol and data siz affcts communication latncy and bandwidth. Transutr links can orat at 5, 10 or 20 MBit/sc link sd. Givn th 11 bit r byt (2 start bit, 8 data bits, 1 sto bit) srial transmission mthod of th links [7], th idal transfr rat is v / 11 MByt/sc, whr v dnots th sd of th givn link. Tabl 1 lists th idal transfr rat valus for uni-dirctional and bi-dirctional data transfr at diffrnt link sds as wll as th ral transfr rats that wr obtaind by Inmos s xrimnt [7]. Th Inmos masurmnt was carrid out using th transutr s intrnal mmory, thrfor communication tim in alications using xtrnal mmory is incrasd by th corrsonding xtrnal mmory accss tim. Not that th bi-dirctional rats ar only tims of th uni-dirctional on.

4 Tabl 1. Idal and actual link data transfr rats Link sd Transutr link transfr rats [MByt/sc] [MBit/sc] uni-dirctional bi-dirctional Idal Inmos Idal Inmos In a transutr rogram th hysical link rmains hiddn to th rogrammr, communication is carrid out via namd channls. Dnding on how rocsss ar mad onto rocssors, ths channls can b intrnal or xtrnal channls. Th way data is transfrrd ovr a channl is dfind by rotocols. Occam and th othr aralll languags for th transutr offr a choic of rotocols to b usd for th communication [8]. In an idal systm th transfr tim is xctd to b roortional to th data siz N, and b indndnt of th transmission mthod and rotocol usd. In ral systms, such as transutr ons, this is not th cas. Th ffct of th two most frquntly usd rotocols, th CHAN OF rimitiv.ty and th CHAN OF countd array, on data transfr is xamind. For simlicity rasons only th CHAN OF BYTE and th CHAN OF INT16::[]BYTE rotocols ar discussd hr, sinc th othr tys can b drivd from ths two. Th link data transfr tim is masurd on a two-transutr systm, whr th two transutrs ar connctd by a dirct, v = 20 MBit/sc, link. 3.1 Th CHAN OF BYTE rotocol Th idal transmission tim, T w', of a on word (on word = on byt) mssag ovr a link can b calculatd as T w = 11. Givn a 20 MBit/sc link T v w' rsults in µsc. In ral systms, howvr, th byt rotocol basd transmission tim is largr du to som intrnal rocssing and th rsnc of th acknowldg signal. Furthrmor, thr is th xtra startu tim that must b addd to th transmission tim ach tim a mssag (in this cas a byt) is bing snt ovr th link. Thrfor th transfr tim, T transfr, of N byts of data is not Ttransfr = N Tw, but rathr Ttransfr = N btw + Ts g=t Byt, whr T w is th ral transmission tim of on byt ovr th link and T s is th mssag startu tim. Startu tim, T s can b stimatd from th assmbly quivalnt of th data transfr cod or simly from masuring th tim of snding N byts of data ovr th link. Tabl 2 includs th masurd transfr tims obtaind by transmitting data of four diffrnt sizs, 4, 16, 64 and 256 KByts, ovr an xtrnal and intrnal channl. Th masurmnts wr rformd using a 20 MBit/sc link. From th masurd data, T Byt bcoms µsc. Tabl 2 also lists th transfr rat valus corrsonding to th masurd data transfr tims. As it has bn xctd, snding data ovr a channl in individual byts is vry infficint (transfr rat = MByt/sc) du to th larg startu tim. It is shown that although th intrnal channl is imlmntd by data transfr in mmory, th transfr rat is aroximatly qual to that of th xtrnal channl. Tabl 2. Data transfr tims and bandwidth rsults xtrnal channl intrnal channl Imag siz tim [msc] MByt/sc tim [msc] MByt/sc

5 Tabl 3. Unit transfr tim valus for diffrnt rimitiv tys rimitiv ty T ty [msc] BYTE INT INT 0.01 REAL REAL If instad of th rimitiv ty BYTE, othr tys, such as INT16, INT, REAL32 or REAL64, should b usd, th transmission tim T ty of a rimitiv data ty is incrasd in roortion to th siz of th nw ty in byts. Tabl 3 lists th valus of T ty for th diffrnt rimitiv tys. 3.2 Th CHAN OF INT16::[]BYTE rotocol It is xctd that th countd array rotocol, whr data is snt ovr th link in ackts, is mor fficint sinc th numbr of communication initiations (i.. startus) is rducd to th numbr of ackts. Lt k dnot th siz of a data ackt. Thn w can dfin th ackt transfr tim function, t (k), which givs th tim of transmitting on ackt ovr th link as t bk g = bk + 2gTw + Ts (2) whr th xtra 2 byts in th ackt lngth ar du to th INT16 tag that rcds th data ackt. From this it follows that th total tim, T transfr, of transmitting N byts of data in ackts of lngth k is givn as or by using Eq. (2), as T N = bk + 2g T + T (3) k transfr w s T transfr N = k t b k g (4) Again, T w and T s is stimatd from masurd data. Tabl 4 lists th masurd transfr tim rsults obtaind by transmitting byts of data in varying lngth ackts. Paramtrs T w and T s can b calculatd from ths data using last squars curv fitting. This rsults in T w = µsc and T s = µsc (at 20 MBit/sc link sd). It can b sn that for small ackt sizs th countd array rotocol is also vry infficint du to th alrady mntiond startu tim and that for small ackt sizs th xtra 2 byts of th INT16 rfix bcoms a larg ovrhad. Howvr, for larg ackt sizs this mthod bcoms vry fficint, almost achiving th bst transfr rat givn by Inmos. Figur 1 shows th lot of th masurd transfr tim valus in function of th ackt lngth, as wll as Eq. (3) fittd onto ths data oints. Tabl 4. Transfr tims for snding N byts of data in varying siz ackts ackt siz ,024 2,048 4,096 8,192 16,384 tim [msc]

6 250 masurd data 200 t (k)=(n/k)[(k+2)t w +T s ] ] c s m[ mit k = Packt lngth, k Figur 1. Data transfr tim in function of ackt siz Th masurd transfr rat varis from to MByt/sc, dnding on th ackt lngth usd for th data transfr. Tabl 4 and Fig. 1 indicat that th ackt basd transfr mthod is fficint for ackt lngth largr than 64. For larg ackts th startu tim bcoms ngligibl comard to th tim of transfrring th data in th ackt. Again, using othr rimitiv tys, such as INT16, INT, REAL32 or REAL64 tys, transfr tim will b altrd by th siz of th nw ty xrssd in byts. Tabl 5. Unit transfr tim and startu tim valus for diffrnt rimitiv tys rimitiv ty Tw [msc] Ts [msc] BYTE INT INT REAL REAL Ovrhad Calculation This sction xamins th aralll imlmntations of th slctd imag rocssing orations, addition, convolution and histogram calculation, in ordr to dtrmin thir communication attrns and basd on this to dvlo fficint communication mthods that minimis th communication ovrhad. Finally an ovrhad modl is dfind for ach cas, which will b usd in th rformanc rdiction Addition: No communication Th addition oration is th idal on for aralll imlmntation. Each rocssor holds on N / sgmnt of th imag of siz N in its local mmory. Sinc th addition is a strict oint oration, ach rocssor can calculat th nw ixl valu indndntly of th othrs, thrfor thr is no ovrhad involvd in th oration, thus th ovrhad function is T o = 0.

7 4.2. Convolution: Narst-nighbour communication Sinc th convolution oration dnds on a small nighbourhood of th givn ixl, whn rforming calculations on th dgs of ths sgmnts, th rocssor rquirs thos ixls, which blong to a nighbour rocssor s sgmnt. Th common ractic is to stor a largr, ovrlad sgmnt on ach rocssor that includs th ixls of th boundary rgion of th nighbour sgmnt. This way th comlt convolution can b rformd locally on ach rocssor, only th bordr valus must b xchangd. This bordr udat oration (bordr swa) must b rformd bfor th convolution is calculatd On-dimnsional array imlmntation Figur 2 illustrats a sgmnt of a rocssor on-dimnsional array with its bordr aras. Th imag siz is N N, th sgmnt siz is N / N. Th swa rocss can b carrid out in two diffrnt ways. On mthod is to swa th bordr aras aftr on anothr (squntial swa). Th othr mthod is to carry out all swas at th sam tim by making us of th bi-dirctional communication rorty of th transutr links (aralll swa). A N B N b b C D Figur 2. Imag sgmnt and its bordrs on a 1-D array Squntial swa Th swa of on b N siz ara is gnrally carrid out by snding b ackts with lngth k = N to th nighbour rocssor. Thus, th swa tim of on ara is b t N j. If ach rocssor rforms th xchang of th four aras squntially, du to th synchronisation ffct, th total tim will b roortional to th lngth of th array. Furthrmor, sinc ach link must accommodat two swa orations, th total swa tim is T = 2b 1 t N swa b g j (5) Th comlxity of th squntial swa oration is Θ(). Howvr, w can gratly imrov th rformanc of th swa if w us th links in aralll, ach rforming a bi-dirctional transfr as it is dscribd nxt. Paralll swa In th aralll imlmntation of th swa oration th to (A, B) and bottom (C, D) aras can b xchangd simultanously. Furthrmor, ach link can accommodat data transmission in both dirctions, thus th snd and rciv orations can b rformd at th sam tim. As th transfr tim in th bi-dirctional mod is a. 1.4 tims of th unidirctional on (s Tabl 1), th total aralll swa tim is givn as

8 T = 1. 4b t N j (6) swa Using this mthod, th swa oration has bcom a constant tim oration and is indndnt of th ntwork siz, Two-dimnsional array imlmntation Whn a two-dimnsional rocssor array is usd, thr ar four bordrs to considr and also sgmnt sizs ar slightly diffrnt from th on-dimnsional array cas, as it is shown in Figur 3. Sinc th squntial imlmntation of th swa oration is vry infficint, hr w considr th 2-D xtnsion of th aralll swa only. Th sgmnt is of siz N / N /, th siz of th outr bordr aras is b N / + 2bj and th siz of th innr bordrs is b N /. To simlify th imlmntation of th swa oration w will us b N / + 2b j bordr siz for th innr aras as wll. b A B E F N G H b C D Figur 3. Imag sgmnt and its bordrs on a 2-D rocssor array Th transutr links would allow th aralll xchang of all th four boundaris simultanously, but bcaus of th cornrs of th outr bordr aras, which blong to diagonally nighbouring rocssors, th swa must b rformd in two sts: a horizontal and a vrtical swa. Using a ackt of lngth k = N / + 2 b for th xchang, th total twodimnsional swa tim is givn by N T = 2. 8b t + 2bj (7) swa Now th swa oration is a dcrasing function of. It coms from Eq. (2) that T swa has a lowr bound of 2.8t (3), thrfor T swa msc Histogram calculation: global communication During histogram calculation th rocssors calculat th local histogram of thir local imag sgmnts in aralll. Ths local rsults ar snt to a mastr rocss to calculat th global histogram. This global histogram is thn snt back (broadcast) to ach nod to mak furthr histogram-basd comutation ossibl. Th algorithm is thus mad u of calculation, gathr and broadcast sts.

9 On-dimnsional array imlmntation In gathr and broadcast basd orations data must travl along th links of th array to rach its dstination. Th tim duration of this travl is roortional to th diamtr of th ntwork. In a on-dimnsional array this is d = - 1. Assuming a 256 gray lvl imag, th total ovrhad of th histogram calculation on a 1-D array is T = b 1gd 2 t b256g + 256t i (8) o INT c whr t INT (256) is th ackt transfr function using INT data ty and t c is th cost of th addition oration Two-dimnsional array imlmntation A squar array roducs similar rsult. Th only diffrnc is in th diamtr of th ntwork, d = 2 1j. Thus, th total ovrhad in th 2-D cas is T = 2 1jd2 t b256g + 256t i (9) o INT c 5. Prdiction Rsults 5.1. Excution tim and sdu rdiction Using th abov dvlod ovrhad functions and th srial xcution tim valus, T 1, listd in Tabl 6, th aralll xcution tim, sdu and fficincy can b rdictd for an arbitrary siz ntwork. To rsrv sac, only th xcution tim rsults for a 1-D array ar shown blow; sdu and fficincy can b drivd from this information. Convolution Figur 4 illustrats th rdictd and masurd xcution tim for th convolution oration imlmntd on a on-dimnsional transutr array. Whn using th squntial swa mthod, th ovrhad is incrasing with and communication tim will dominat xcution tim for N / 2, and consquntly th xcution tim will incras and sdu will start to dcras. Th aralll swa mthod roducs a constant tim ovrhad function, thrfor th xcution tim in this cas will dcras monotonly for vry good scalability. Tabl 6. Srial xcution tims for diffrnt imag sizs imag Srial xcution tim, T1 [msc] siz addition conv 3 3 conv 5 5 conv 7 7 histogram N, rsulting in

10 ] c s , b=3 5 5, b=2 3 3, b=1 rdictd, squntial swa rdictd, aralll swa masurd m[ ) ( mit g ol , b=3 5 5, b=2 3 3, b= log (rocssors) Figur 4. Prdictd and masurd xcution tim for convolution on a 1-D array Th two-dimnsional array imlmntation of th convolution oration dislays vn bttr rformanc, as th swa oration is a dcrasing function of. Thrfor th xcution tim is monotonly dcrasing until N and consquntly th scalability of th algorithm is idal. Th accuracy of th rdiction mthod is shown in Tabl 7. Th largst diffrnc is %, whil th avrag diffrnc is lss than 0.16 %. Tabl 7. Th rror of xcution tim rdiction for th convolution oration Prdiction rror for convolution [%] Histogram Th rdictd and masurd xcution tims of th histogram oration ar shown in Figur 5 for a on-dimnsional array. Th ffct of th incrasing ovrhad function (Eq. 8) is obvious. Communication tim will vry soon start to dominat and aftr raching th balanc oint incras xcution tim. It can b sn also that th imlmntation is snsitiv for th roblm siz. For small imag sizs (.g ) thr is almost no bnfit from using aralllism, whil for larg imags ( or largr) w can gain rformanc incras u to rocssors.

11 ] c s m[ ) ( mit g ol rdiction data masurd data log (rocssors) Figur 5. Prdictd and masurd xcution tim for th histogram oration on a 1-D array Th accuracy of th rdiction mthod is shown in Tabl 8. Th largst diffrnc is %, whil th avrag diffrnc is lss than 0.75 %. Th accuracy of our analytical mthod is notabl whn comard to rsults of othr rformanc rdiction mthods [1][6]. Tabl 8. Th rror of xcution tim rdiction for th histogram oration Prdiction rror for histogram [%] Dtcting rformanc-critical oints, imroving th imlmntation Th dscribd analytical modl nabls th rogrammr or systm dsignr to invstigat th ffcts of algorithmic and/or systm aramtr changs on th rformanc. This can b vry usful in th rformanc tuning or in th architctur slction rocss. Our modl dnds on th following aramtrs: t c (imlicit in T 1 ), T s mssag startu tim, communication sd v (imlicit in T w ), th diffrnt roblm dcomositions ar xrssd in th communication attrn, thus in th communication ovrhad. Th ffct of changs in ths aramtrs can b xamind and asily visualisd with an aroriat lotting rogram. Whil in a strict transutr systm th communication and rocssor sd aramtrs ar constants, ths aramtrs mak it ossibl to comar th rformanc of a givn aralll algorithm on diffrnt mssag assing architcturs (.g. th C40 rocssors or workstation clustrs, tc.).

12 Th modl also hls to dtct th critical oints of th algorithms that rduc rformanc. By analysing th ovrhad function, th sourc of th rformanc limiting trms can b idntifid and by changing th roblm dcomosition, th communication stratgy or th rotocol, th ffct of ths trms can b waknd or liminatd. Convolution W saw that th convolution oration using th aralll swa tchniqu rforms idally on both th on-dimnsional and th two-dimnsional arrays. This rsult was xctd naturally as only narst-nighbour communication was usd. Th swa orations usd th most straightforward communication stratgy, data was bing snt by ackts of lngth qual to on row of a sgmnt. This rsultd in th ovrhad functions of Eq. (6) and Eq. (7). From ths quations it can b sn that if th full amount of data is snt in on ackt, instad of b smallr ackts, th frquncy of communication can b rducd, thrfor th startu tim will hav to b accountd for only onc. This rsults in th following ovrhad functions: T = 1. 4t b N j for th linar rocssor array and swa swa T = 2. 8t b N / + 2bjj for th two-dimnsional array, and in imrovd rformanc consquntly. Histogram W hav sn in Sction 5.1 that th histogram oration bcoms infficint as incrass du to th global communication. Nxt w show that it is ossibl to us mor fficint mthods for communication and this will rsult in imrovd rformanc. Th dominating factors in th ovrhad functions, Eq. (8) and Eq. (9), ar th gathr and broadcast arts. Sinc th broadcast oration snds th sam data to all rocssors, this tim can b rducd by using ilining tchniqu during th transmission [9]. If w can divid th data N into smallr ackts of lngth k and dnot th distanc to b travlld by th mssag with m, thn th broadcast tim is givn as N T = + m 1 t k broadcast c h b g k (10) If k = 1, thn Eq. (10) bcoms b N + m 1gt b1g. In this cas a byt buffr and individual byt transfr can b usd instad of th countd array rotocol, rsulting in T = b N + m 1g T (11) broadcast Eq. (10) is minimisd at diffrnt valus of k for diffrnt distancs. It is shown in [10] that th otimal broadcast mthod giving th shortst communication tim is dscribd by th following function: T b broadcast m g = Byt + m 1 t 32 if 1< m 3 R 256 c h b g 32 INT 256 c h b g 16 INT 256 Sc h b g 8 INT 256 c h b g 4 INT b T + m 1 t 16 if 3 < m 7 + m 1 t 8 if 7 < m 63 + m 1 t 4 if 63 < m m 1gT INT if 127 < m (12)

13 ] c s Using this imrovd broadcast mthod and a diffrnt communication and rocss allocation stratgy, th rformanc of th histogram calculation can b imrovd. Th basic ida of th suggstd modification is that instad of th traditional imlmntation of th gathr rocss, whr ach rocssor is waiting for its nighbours and du to this synchronisation th histogram rocss bcoms squntial, w try to artition th ntwork in a way that th combining rocsss could b xcutd in aralll. With this artitioning th combining of th local histograms can b rformd in log 2 tim sts instad of th -1 sts of th traditional mthod [10]. Alying Eq. (8) and (9) on ths rducd diamtr arrays, th total ovrhad in th histogram oration rsults in th followings for a 1-D and 2-D array, rsctivly: T F I = t b256g t log + T G J INT 2 2 H (13) K o c broadcast 1 To = t b 256g + 256tc log + T broadcast 1j INT 2 (14) 2 Th rformanc achivd with this imrovd histogram algorithm is comard to th traditional and th idal tr-connctd ntwork imlmntation in Figur 6. It is shown that th rformanc has bn imrovd with ths modifications and th rang of th fficint aralll oration has bn xtndd. Unfortunatly, th ovrhad function is still roortional to, thrfor th communication tim will again limit th achivabl sdu. Obviously, if th givn transutr systm can b rconfigurd dynamically into a binary tr toology, th bst rformanc can b achivd. In this ar howvr, a static toology has bn assumd traditional histogram imrovd histogram histogram on tr toology m[ ) ( mit g ol log (rocssors) Figur 6. Comarison of th rformanc of diffrnt histogram algorithms

14 6. Conclusions and Futur Work This ar rsntd an analytical rformanc rdiction mthod for transutr basd lowlvl, aralll imag rocssing orations. Th slctd orations rrsnt th algorithm classs with no, local and global communication. Th analytical modl is basd on th fixd siz sdu formula and uss a global ovrhad function to xrss th ffcts of comutation, communication ovrhad, synchronisation dlays and hardwar aramtrs. Sinc communication is a dominant factor of th ovrhad, th ffct of communication on th rformanc is xamind. Two frquntly usd communication rotocols hav bn analysd and link data transfr modls hav bn drivd for both. Ths modls ar usd to rrsnt hardwar aramtrs in th ovrhad function. Th nar-nighbour communication class convolution and th gathr-broadcast class histogram algorithms hav bn xamind on a on-dimnsional and on a two-dimnsional transutr array and aftr analysing thir communication attrns th ovrhad function for ach oration has bn dvlod. Exrimntal data was obtaind to comar th validity of th link transfr modls and th ovrhad functions. All masurmnts hav bn carrid out on a 16-nod transutr systm quid with T800 rocssors. Th ovrhad functions wr usd to rdict th xcution tim of th givn algorithmmachin combination. Th sdu and scalability mtrics can b drivd from th aralll xcution tim. Th ffct of systm aramtr changs on rformanc can b studid with th modl. It has bn shown that by idntifying rformanc critical arts of th algorithm, it is oftn ossibl to imrov rformanc by a chang in th communication stratgy. Th mthod rovids accurat rdiction and can hl rogrammrs, systm dsignr ffctivly to dsign high-rformanc, fficint scalabl algorithms and systms. Futur work would includ th gnralisation of th mthod for othr algorithms and diffrnt hardwar architcturs including th T9000 rocssor, aralll DSP systms and workstation clustrs. Acknowldgmnt This work was suortd in art by th Hungarian National Scinc Foundation (OTKA) undr Grant F Rfrncs [1] Mark J. Clmnt and Michal J. Quinn, Analytical Prformanc Prdiction on Multicomutrs, in Procdings Surcomuting '93, [2] E. Glnb, Multirocssor Prformanc, John Wily & Sons, [3] A. Y. Grama, A. Guta and V. Kumar, Isofficincy: Masuring th Scalability of Paralll Algorithms and Architcturs, IEEE Paralll and Distributd Tchnology, Vol. 1, No. 3, 12-21, August [4] X. Zhang, Y. Yan, Q. Ma, Masuring and Analyzing Paralll Comuting Scalability, in Proc Int. Conf. on Paralll Procssing, CRC Prss, August [5] Mark E. Crovlla and Thomas J. LBlanc, Th Sarch for Lost Cycls: A Nw Aroach to Paralll Program Prformanc Evaluation, Tchnical Rort 479, Th Univrsity of Rochstr, Dcmbr [6] M. J. Zmrly t al., Charactrising Comutational Krnls to Prdict Prformanc of Paralll Systms, in A. D Gloria t al. (Eds.), Transutr Alications and Systms '94, , IOS Prss, [7] Inmos Ltd., Th Transutr Databook, Prntic Hall, [8] Inmos Ltd., occam 2 Rfrnc Manual, Prntic Hall, [9] P. Brtskas, J. N. Tsitsiklis, Paralll and Distributd Comutation, Prntic Hall, [10] Z. Juhasz, Efficint Communication Mthods for Minimising Ovrhad in Paralll Imag Procssing Algorithms, in Proc. 2nd Austrian-Hungarian Worksho on Transutr Alications, 1994.